On the well-posedness of the stochastic Allen-Cahn equation in two dimensions. (English) Zbl 1242.65013
Summary: White noise-driven nonlinear stochastic partial differential equations (SPDEs) of parabolic type are frequently used to model physical systems in space dimensions \(d = 1, 2, 3\). Whereas existence and uniqueness of weak solutions to these equations are well established in one dimension, the situation is different for \(d \geqslant 2\). Despite their popularity in the applied sciences, higher dimensional versions of these SPDE models are generally assumed to be ill-posed by the mathematics community. We study this discrepancy on the specific example of the two dimensional Allen - Cahn equation driven by additive white noise. Since it is unclear how to define the notion of a weak solution to this equation, we regularize the noise and introduce a family of approximations. Based on heuristic arguments and numerical experiments, we conjecture that these approximations exhibit divergent behavior in the continuum limit. The results strongly suggest that shrinking the mesh size in simulations of the two-dimensional white noise-driven Allen - Cahn equation does not lead to the recovery of a physically meaningful limit.
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 35Q35 | PDEs in connection with fluid mechanics |
Keywords:
well-posedness; Allen-Cahn equation; white noise; nonlinear stochastic partial differential equations of parabolic type; numerical experimentsReferences:
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